{
  "id": "1911.02763",
  "version": "v1",
  "published": "2019-11-07T05:47:11.000Z",
  "updated": "2019-11-07T05:47:11.000Z",
  "title": "On a New Graph defined on the order of elements of a Finite Group",
  "authors": [
    "Subarsha Banerjee"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper, the author has introduced a new graph structure called the \\textbf{Co-Prime Order Graph} $\\Theta(G)$ on a finite group $G$ whose vertex set is $G$ and any two vertexes $x,y$ in $\\Theta(G)$ are adjacent if and only if $\\gcd(o(x),o(y))=1$ or prime. We study how the graph properties of $\\Theta(G)$ and group properties of $G$ are related among themselves. We have given various conditions when $\\Theta(G)$ is connected, complete, planar and Hamiltonian when $G=\\Bbb Z_n$ and $G=D_n$ for various $n\\in \\Bbb N$. We have also found when the graph $\\Theta(G)$ is Eulerian for any finite group $G$. We have also studied the vertex connectivity of $\\Theta(\\Bbb Z_n)$ for various $n\\in \\Bbb N.$ Finally we have computed the Signless Laplacian Spectra of $\\Theta(G)$ when $G=\\Bbb Z_n$ and $G=D_n$ for $n=pq$ and $n=p^m$ where $p,q$ are primes and $m\\in \\Bbb Z$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-07T05:47:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C25",
      "05C50"
    ],
    "keywords": [
      "finite group",
      "graph structure",
      "order graph",
      "graph properties",
      "group properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}